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If a tangent of slope 2 of the ellipse (...

If a tangent of slope 2 of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` is normal to the circle `x^2+y^2+4x+1=0` , then the maximum value of `a b` is 4 (b) 2 (c) 1 (d) none of these

A

4

B

2

C

1

D

none of these

Text Solution

Verified by Experts

A tangent of slopw 2 to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` is `y==4x+-sqrt(4a^(2)+b^(2))" "(1)`
This is normal to the circle `x^(2)+y^(2)+4x+1=0`
Therefore, (1) passes througj (-2,0), i.e.,
or `0=-4+-sqrt(4a^(2)+b^(2))`
or `4a^(2)+b^(2)=16`
Using `AMgeGM`, we get
`(4a^(2)+b^(2))/(2)gesqrt(4a^(2)-b^(2))`
or `able4`
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